# Graduate dissertations

Title: K-Theory Correspondences and the Fourier-Mukai Transform

Speaker: Daniel Hudson, University of Victoria

Date and time:
26 Apr 2019,
10:30am -
11:30am

Location: Clearihue C118

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Notice of the Final Oral Examination for the Degree of Master of Science
of

DANIEL HUDSON

BSc (University of Victoria, 2016)

“K-Theory Correspondences and the Fourier-Mukai Transform”

Department of Mathematics and Statistics

Friday, April 26, 2019 10:30 A.M.

Clearihue Building Room C118

Supervisory Committee:

Dr. Heath Emerson, Department of Mathematics and Statistics, University of Victoria (Co-Supervisor)

Dr. Ian Putnam, Department of Mathematics and Statistics, UVic (Co-Supervisor)

External Examiner:

Dr. Rufus Willett, Department of Mathematics, University of Hawaii

Chair of Oral Examination:

Dr. Sara Ramshaw, Faculty of Law, UVic

Abstract

The goal of this thesis is to give an introduction to the geometric picture of bivariant K- theory developed by Emerson and Meyer building on the ideas Connes and Skandalis, and then applying the machinery to prove a result of Emerson stated, but not proved, in a recent article [8]. We begin by giving an overview of topological K-theory, necessary for developing bivariant K-theory. Then we discuss Kasparov’s analytic bivariant K-theory, and from there develop topological bivariant K-theory. In the final chapter we state and prove the result of Emerson.

Title: Population trends in golden eagle migration, Mount Lorette, Alberta, 1992 - 2018

Speaker: Ben Liu, University of Victoria

Date and time:
26 Apr 2019,
10:00am -
11:00am

Location: Clearihue C116

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The Final Oral Examination for the Degree of

Master of Science

(Department of Mathematics and Statistics)

Ben Liu

2016 East China Normal University B.Sc.

Population trends in golden eagle migration, Mount Lorette, Alberta, 1992 - 2018

Friday, April 26th, 2019 10:00am CLEARIHUE BLDG C116

Supervisory Committee:

Dr. Laura Cowen, Department of Mathematics and Statistics, UVic (Supervisor)

Dr. Julie Zhou, Department of Mathematics and Statistics, UVic (Member)

Chair of Oral Examination:

Dr. Christopher Eagle, Department of Mathematics and Statistics, UVic

Title: Statistical Methods for Neuroimaging Data Analysis and Cognitive Science

Speaker: Yin Song, University of Victoria

Date and time:
24 Apr 2019,
1:00pm -
2:00pm

Location: Clearihue B007

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Notice of the Final Oral Examination for the Degree of Doctor of Philosophy
of

YIN SONG

MSc (University of Alaska Fairbanks, 2013) BSc (Northwest A&F University, 2010)

“Statistical Methods for Neuroimaging Data Analysis and Cognitive Science”

Department of Mathematics and Statistics

Wednesday, April 24, 2019 1:00 P.M.

Clearihue Building Room B007

Supervisory Committee:

Dr. Farouk Nathoo, Department of Mathematics and Statistics, University of Victoria (Supervisor)

Dr. Laura Cowen, Department of Mathematics and Statistics, UVic (Member)

Dr. Michael Masson, Department of Psychology, UVic (Outside Member)

External Examiner:

Dr. Marc Fredette, Decision Science, HEC Montreal

Chair of Oral Examination:

Dr. Cornelis van Kooten, Department of Economics, UVic

Abstract

This thesis presents research focused on developing statistical methods with emphasis on tools that can be used for the analysis of data in neuroimaging studies and cognitive science. The first contribution addresses the problem of determining the location and dynamics of brain activity when electromagnetic signals are collected using magnetoencephalography (MEG) and electroencephalography (EEG). We formulate a new spatiotemporal model that jointly models MEG and EEG data as a function of unobserved neuronal activation. To fit this model we derive an efficient procedure for simultaneous point estimation and model selection based on the iterated conditional modes algorithm combined with local polynomial smoothing. The methodology is evaluated through extensive simulation studies and an application examining the visual response to scrambled faces.

In the second contribution we develop a Bayesian spatial model for imaging genetics developed for analyses examining the influence of genetics on brain structure as measured by MRI. We extend the recently developed regression model of Greenlaw et al. (Bioinformatics, 2017) to accommodate more realistic correlation structures typically seen in structural brain imaging data. We allow for spatial correlation in the imaging phenotypes obtained from neighbouring regions in the same hemisphere of the brain and we also allow for correlation in the same phenotypes obtained from different hemispheres (left/right) of the brain. This correlation structure is incorporated through the use of a bivariate conditional autoregressive spatial model. Both Markov chain Monte Carlo (MCMC) and variational Bayes approaches are developed to approximate the posterior distribution and Bayesian false discovery rate (FDR) procedures are developed to select SNPs using the posterior distribution while accounting for multiplicity. The methodology is evaluated through an analysis of MRI and genetic data obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) and we show that the new spatial model exhibits improved performance on real data when compared to the non-spatial model of Greenlaw et al. (2017).

In the third and final contribution we develop and investigate tools for the analysis of binary data arising from repeated measures designs. We propose a Bayesian approach for the mixed-effects analysis of accuracy studies using mixed binomial regression models and we investigate techniques for model selection. Software and examples in R and JAGS for implementing the proposed analysis are described and available at https://v2south.github.io/BinBayes/.

Title: Mixing Weight Estimation in Poisson Mixture Models

Speaker: Yongchen Huang, University of Victoria

Date and time:
12 Apr 2019,
3:30pm -
4:20pm

Location: Clearihue B021

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The Final Oral Examination for the Degree of
Master of Science (Department of Mathematics and Statistics)

Yongchen Huang

2015 University of Chongqing B.Sc.

Mixing Weight Estimation in Poisson Mixture Models

Friday, April 12th, 2019 3:30pm

CLEARIHUE BLDG B021

Supervisory Committee:

Dr. Mary Lesperance, Department of Mathematics and Statistics, UVic (Supervisor)

Dr. Min Tsao, Department of Mathematics and Statistics, UVic (Member)

Chair of Oral Examination:

Dr. Kieka Mynhardt, Department of Mathematics and Statistics, UVic

Title: Ramp Approximations of Finitely Steep Sigmoid Control Functions in Soft-Switching ODE Networks

Speaker: Graham Quee

Date and time:
11 Apr 2019,
9:30am -
10:20am

Location: Clearihue B019

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Notice of the Final Oral Examination for the Degree of Master of Science

of

GRAHAM QUEE

BSc (University of Victoria, 2017)

“Ramp Approximations of Finitely Steep Sigmoid Control Functions in Soft-Switching ODE Networks”

Department of Mathematics and Statistics

Thursday, April 11, 2019 9:30 A.M.

Clearihue Building Room B019

Supervisory Committee:

Dr. Roderick Edwards, Department of Mathematics and Statistics, University of Victoria (Supervisor)

Dr. Junling Ma, Department of Mathematics and Statistics, UVic (Member)

External Examiner:

Dr. Tomas Gedeon, Mathematical Sciences, Montana State University

Chair of Oral Examination:

Dr. Michael McGuire, Department of Electrical and Computer Engineering, UVic

Abstract
In models for networks of regulatory interactions in biological molecules, the sigmoid relationship between concentration of regulating bodies and the production rates they control has lead to the use of continuous time ‘switching’ ordinary differential equations (ODEs), sometimes referred to as Glass networks. These Glass networks are the result of a simplifying assumption that the switching behaviour occurs instantaneously at particular threshold values. Though this assumption produces highly tractable models, it also causes analytic difficulties in certain cases due to the discontinuities of the system, such as non-uniqueness. In this thesis we explore the use of ‘ramp’ functions as an alternative approximation to the sigmoid, which restores continuity to the ODE and removes the assumption of infinitely fast switching by linearly interpolating the focal point values used in a corresponding Glass network. A general framework for producing a ramp system from a certain Glass network is given. Solutions are explored in two dimensions, and then in higher dimensions under two different restrictions. Periodic behaviour is explored in both cases using mappings between threshold boundaries. Limitations in these method are explored, and a general proof of the existence of periodic solutions in negative feedback loops is given.